Cross product with orthonormal basisIt does if you meant to write $(f_{u_1}(p),f_{u_2}(p),f_{u_3}(p))\times(g_{u_1}(p),g_{u_2}(p),g_{u_3}(p))$. Then you're taking the cross product of the same two vectors on each side of the equation.

How do we represent this event?The CDF is the probability that $M$ is any value up to $x$: $P(M < x) = 1 - P(M\geq x) = 1 - (1-x)^2$. The PDF is the derivative of the CDF: $d(1-(1-x)^2)/dx = 0 - 2(1-x)^1 (-1) = 2(1-x)$.

Nov19

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How do we represent this event?Both -- the area in the plane is the probability of the corresponding event, since $X$ and $Y$ are independent and uniform. I've edited to elaborate.